Positive solutions of positive linear systems
type: Item
instance of: scholarly article
title: Positive solutions of positive linear systems
zbMATH Open document ID: 0565.34008
DOI: 10.1016/0024-3795(85)90271-X
author: J. Blot
published in: Linear Algebra and its Applications
publication date: 1985-01-01T00:00:00Z
review text: The Volterra-Lotka equations \(dx_ i/dt=x_ i(b_ i-\Sigma a_{ij}x_ j),\) \(b_ i>0\), \(a_{ii}>0\), \(a_{ij}\geq 0\) correspond to an ecosystem with interspecies and intraspecies competition. A steady state solution \(X^*\) exists if \(b_ i-\Sigma a_{ij}X^*_ j=0\). The author shows that A is invertible with \(A^{-1}b\) having all positive elements, so that such a stable \(X^*\) exists, provided \[ b_ i>\sum^{n}_{j=1,j\neq i}a_{ij}b_ j/a_{jj},\quad i=1,...,n. \] An iterative method of approximating \(X^*\) is given.
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 3901176
zbMATH Keywords: linear nonhomogeneous system
zbMATH Keywords: Volterra-Lotka equations
zbMATH Keywords: ecosystem
zbMATH Keywords: steady state solution
zbMATH Keywords: iterative method
MaRDI profile type: Publication
cites work: Q3943082
cites work: Time delays in n-species competition – I: Global stability in constant environments
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description: scientific article; zbMATH DE number 3901176